1
$\begingroup$

Let $X_{1}, \dots, X_{n}$ be i.i.d. ramdom variables having mean $\mu$ and standard deviation $\sigma$. I wonder if the "studentized" $X_{i}$, the sample version of standardized $X_{i}$ where $\mu$ is replaced with the sample average and $\sigma$ is replaced with a sample standard deviation, admits a relatively canonical terminology in literature?

In my shallow opinion, the term "studentized" is informative but would cost some possibilities of confusion. So a terminology, if it exists, is sought.

The following is in response to a question raised in a comment below. To avoid introducing too many symbols, I described it in plain English. If this helps: If $$ \frac{X_{i} - \mu}{\sigma} $$ is called the standardized $X_{i}$, if $\bar{X}$ denotes the sample average, and if $s$ denotes the sample standard deviation under consideration, then I call $$ \frac{X_{i} - \bar{X}}{s} $$ the studentized $X_{i}$ above for reference ease.

Improve this question
$\endgroup$
4
  • $\begingroup$ Can you give a specific example, to explain what you exactly mean with these 'replacements', and with 'sample version of a variable $X_i $'. (note the value $\bar x / s_x $ is called the t-score, which contrasts with z-score, did you think about something like that?) $\endgroup$
    – Sextus Empiricus
    Commented Nov 25, 2018 at 16:44
  • $\begingroup$ @MartijnWeterings, Update is available. $\endgroup$
    – Yes
    Commented Nov 25, 2018 at 16:53
  • 1
    $\begingroup$ I am still a bit confused. Is this about computing a z-statistic or t-statistic (as used in z-test and t-test), or is this about normalization (I notice now you explicitly used the term 'standardized')? What confuses me (and the formula's don't solve this) are the random variables $X_1, ... , X_n$. Is $X_i$ a random variable (e.g. $X_i$ is the length of a person from population $i$) or is $X_i$ a specific member of a sample (e.g. $X_i$ is the length of person $i$ in the sample)? $\endgroup$
    – Sextus Empiricus
    Commented Nov 29, 2018 at 12:48
  • $\begingroup$ A new thing: it might be also useful to give examples/explanation where/how you wish to apply the term 'studentized'? If you use it in some context of defining confidence intervals (or anything else where distinction between the distributions normal/t is important) then it might be useful (e.g. see studentized residuals). If you use it a context like pca where one just seeks to roughly normalize the vectors then it will be confusing. $\endgroup$
    – Sextus Empiricus
    Commented Nov 29, 2018 at 12:59

1 Answer 1

Reset to default
1
$\begingroup$

Some of the comments in the original post are worthy as an answer. I summarize as follows.

Original post asks whether $\dfrac{x_i-\bar{x}}{s}$ can be referred to as "studentized value of $x_i$"; since $\dfrac{X_i-\mu}{\sigma}$ is referred to as "standardized value of $X_i$". The names are referring to Student's t and standard normal distribution, respectively.

"Studentized" in this case may as well just be called just "standardized", unless in cases where distinction between t-distribution and normal distribution is important e.g. studentized residuals. Most of the time it will just add confusion and it's better to just use the term "standardized".

Improve this answer
$\endgroup$
1
  • $\begingroup$ +1 I typically say "Standardized using the sample mean and sample standard deviation." Depending on the audience I might say "z transformation" or "t transformation" to convey the distinction. $\endgroup$
    – Alexis
    Commented Feb 3, 2022 at 16:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.